Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


On the location of critical points of polynomials

Authors: Branko Curgus and Vania Mascioni
Journal: Proc. Amer. Math. Soc. 131 (2003), 253-264
MSC (2000): Primary 30C15; Secondary 26C10
Published electronically: June 3, 2002
MathSciNet review: 1929045
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given a polynomial $p$ of degree $n \geq 2$ and with at least two distinct roots let $Z(p) = \{z : p(z) = 0\}$. For a fixed root $\alpha \in Z(p)$ we define the quantities $\omega(p,\alpha) := \min\bigl\{\vert\alpha - v\vert : v \in Z(p)\setminus \{\alpha\} \bigr\}$and $\tau(p,\alpha) := \min\bigl\{\vert\alpha - v\vert : v \in Z(p')\setminus \{\alpha\} \bigr\}$. We also define $\omega(p)$ and $\tau(p)$ to be the corresponding minima of $\omega(p,\alpha)$ and $\tau(p,\alpha)$ as $\alpha$ runs over $Z(p)$. Our main results show that the ratios $\tau(p,\alpha)/\omega(p,\alpha)$ and $\tau(p)/\omega(p)$ are bounded above and below by constants that only depend on the degree of $p$. In particular, we prove that $(1/n)\omega(p)\leq\tau(p)\leq\bigl(1/2\sin(\pi/n)\bigr)\omega(p)$, for any polynomial of degree $n$.

References [Enhancements On Off] (What's this?)

  • [1] Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960 (97e:41001)
  • [2] Peter Henrici, Applied and computational complex analysis. Vol. 1, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Power series—integration—conformal mapping—location of zeros; Reprint of the 1974 original; A Wiley-Interscience Publication. MR 1008928 (90d:30002)
  • [3] Morris Marden, Geometry of polynomials, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972 (37 #1562)
  • [4] Maurice Mignotte, Mathematics for computer algebra, Springer-Verlag, New York, 1992. Translated from the French by Catherine Mignotte. MR 1140923 (92i:68071)
  • [5] A. M. Ostrowski, On the moduli of zeros of derivatives of polynomials, J. Reine Angew. Math. 230 (1968), 40–50. MR 0225973 (37 #1563)
  • [6] J. L. Walsh, On the location of the roots of the derivative of a polynomial, C. R. Congr. Internat. des Mathématiciens, pp. 339-342, Strasbourg, 1920.
  • [7] J. L. Walsh, The Location of Critical Points of Analytic and Harmonic Functions, American Mathematical Society Colloquium Publications, Vol. 34, American Mathematical Society, New York, N. Y., 1950. MR 0037350 (12,249d)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 30C15, 26C10

Retrieve articles in all journals with MSC (2000): 30C15, 26C10

Additional Information

Branko Curgus
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225

Vania Mascioni
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225

PII: S 0002-9939(02)06534-6
Keywords: Roots of polynomials, critical points of polynomials, separation of roots
Received by editor(s): July 10, 2001
Received by editor(s) in revised form: September 4, 2001
Published electronically: June 3, 2002
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2002 American Mathematical Society

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia